The document discusses reference frame theory for modeling electrical machines. It describes how reference frame transformations were developed to simplify differential equations for machines by reducing complexity. Specifically, it outlines transformations developed by Park, Stanley, Kron, Brereton, Krause/Thomas, and Clarke for modeling synchronous and induction machines using rotating reference frames. The advantages of these transformations are that they reduce the number of equations, make coefficients time-invariant, and allow easier analysis and control implementation.

1. UNIT- II
Reference frame theory
16 November 2012 1PRB/Dept.of EEE/SCE

2. INTRODUCTION
Generally, the machine model can be described by differential equations with
time varying mutual inductances,
For example
( for a synchronous machine) (for a induction machine)
but such a model tends to be very complex under dynamic analysis due to
the following reasons
(1) electric circuit in relative motion
(2) electric circuits with varying magnetic reluctance
So that, Several change of variables are used to reduce the complexity of
these differential equations.
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3. TRANSFORMATION
The process of replacing one set of variables by another related set of variables
is called transformation. In the study of power systems and electrical machine
analysis, mathematical transformations are often used to decouple variables, to
facilitate the solution of difficult equations with time-varying coefficients, or to
refer all variables to a common reference frame.
Evolution of transformation technique
R. H. Park - Transformation – 1920 ( for a synchronous machine)
He transformed a change of variables (voltages ,currents and flux linkages)
associated with the stator windings of a synchronous machine to a variables
associated with fictious windings rotating with the rotor at synchronous
speed.
(The stator variables are transformed to a synchronously rotating reference
frame fixed in the rotor)
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4. H. C. Stanley - Transformation – 1930 (for a induction machine)
He transformed a change of variables associated with the rotor windings(rotor
variables) of a induction machine to a variables associated with fictious
stationary windings (stationary reference frame) fixed in the stator
G. Kron - Transformation (for a induction machine)
He transformed a change of variables associated with both stator and rotor
windings of a symmetrical induction machine to a variable associated with
reference frame rotating in synchronism with the rotating magnetic field. This
reference frame is commonly referred to as the synchronously rotating
reference frame.
D. S . Brereton -Transformation ( for a induction machine)
He proposed a change of variables associated with the stator windings of a
induction machine to a variables associated with the reference frame fixed in
the rotor.
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5. Krause and Thomas- Transformation (after 1965)
Both were shown that time varying inductances can be eliminated by referring
the stator and rotor variables to a common reference frame which may rotate at
any speed. This common reference frame is also called arbitrary reference
frame.
E. Clarke - Transformation
He transformed a change of variables associated with the stationary circuits
to a variables associated with stationary reference frame. The stationary
two-phase variables of Clarke’s transformation are denoted as α and β. (Both
are orthogonal).
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6. Advantages of reference frame transformation
1. The number of voltage equations are reduced.
2. The time – varying voltage equations become time – invariant
ones.
3. Performance of power systems and electric machines can be
analyzed without complexities in the voltage equations.
4. Transformations make it possible for control algorithms to be
implemented on the DSP
5. With aid of this technique , many of the basic concepts and
interpretations of this general transformations are concisely
established.
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